2025/08/29 by Olivia Caramello, Caramello, Olivia, Laurent Lafforgue +1
Mathematics · #03G30 #18B25 #18C10 #Advanced Topology and Set Theory #Algebraic Geometry (math.AG) #Category Theory (math.CT) #FOS: Mathematics #History and Theory of Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO)
paper · pdf · doi:10.48550/arxiv.2508.21609
openalex publication_date 2025/08/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Topos theory occupies a singular place in contemporary mathematics: born from Grothendieck's algebraic geometry, it has emerged as a unifying language for geometry, topology, algebra, and logic. This book offers a progressive introduction that moves from familiar ground - groups and their actions, topological spaces, categories, and sheaves - to Grothendieck topologies and sites, then to the axiomatic and categorical foundations of toposes (via Giraud's theorem), to their geometry (morphisms, points, subtoposes, localizations) and finally to their deep ties with geometric logic through classifying toposes. Aimed at readers with a basic familiarity with algebra, general topology, and category theory, the book emphasizes reversible viewpoints - external/internal, local/global, syntactic/semantic - guided by canonical examples, key theorems, and universal constructions. It equips the reader with a workable language in which spaces become categories of sheaves and theories become places classified by toposes. Future chapters will present "toposes as bridges" and relative toposes.